which is reduced from () via a nonlocal reduction \(v(x,t)=\delta u(-x,-t)\) .
Note that nonlocal integrable systems were first systematically proposed in 2013
in [1]} and have received intensive attention
(e.g.[2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}).
The reduction also enables us to see how the distribution of eigenvalues varies with the constraints
imposed in the local and nonlocal reductions.
It is worthy to mention that, apart from those solutions related to complex discrete eigenvalues
(cf.[17]}, [18]}, [19]}),
the FL equation (REF ) allows solutions related to real discrete eigenvalues.
These solutions exhibit (multi-)periodic behaviors,
and also provide solitary waves with algebraic decay as \(|x|\rightarrow \infty \) ,
which are not found in the analytic approaches based on spectral analysis
(cf.[17]}, [18]}, [19]}).
